140 research outputs found
Graph presentations for moments of noncentral Wishart distributions and their applications
We provide formulas for the moments of the real and complex noncentral
Wishart distributions of general degrees. The obtained formulas for the real
and complex cases are described in terms of the undirected and directed graphs,
respectively. By considering degenerate cases, we give explicit formulas for
the moments of bivariate chi-square distributions and Wishart
distributions by enumerating the graphs. Noting that the Laguerre polynomials
can be considered to be moments of a noncentral chi-square distributions
formally, we demonstrate a combinatorial interpretation of the coefficients of
the Laguerre polynomials
"Tail probabilities of the limiting null distributions of the Anderson-Stephens statistics"
For the purpose of testing the spherical uniformity based on i.i.d. directional data (unit vectors) zi , i =1,...,n, Anderson and Stephens (1972) proposed testing procedures based on the statistics Smax = maxuS (u) and S min = minuS (u), where u is a unit vector and nS (u) is the sum of square of u'zi's. In this paper we also consider another test statistic Srange = Smax |Smin. We provide formulas for the P-values of Smax , Smin , Srange by approximating tail probabilities of the limiting null distributions by means of the tube method, an integral-geometric approach for evaluating tail probability of the maximum of a Gaussian random field. Monte Carlo simulations for examining the accuracy of the approximation and for the power comparison of the statistics are given.
"Maximum Covariance Di erence Test for Equality of Two Covariance Matrices"
We propose a test of equality of two covariance matrices based on the maximum standardized di erence of scalar covariances of two sample covariance matrices.We derive the tail probability of the asymptotic null distribution of the test statistic by the tube method.However the usual formal tube formula has to be suitably modi ed,because in this case the index set, around which the tube s formed,has zero critical radius.
Expected Euler characteristic method for the largest eigenvalue: (Skew-)orthogonal polynomial approach
The expected Euler characteristic (EEC) method is an integral-geometric
method used to approximate the tail probability of the maximum of a random
field on a manifold. Noting that the largest eigenvalue of a real-symmetric or
Hermitian matrix is the maximum of the quadratic form of a unit vector, we
provide EEC approximation formulas for the tail probability of the largest
eigenvalue of orthogonally invariant random matrices of a large class. For this
purpose, we propose a version of a skew-orthogonal polynomial by adding a side
condition such that it is uniquely defined, and describe the EEC formulas in
terms of the (skew-)orthogonal polynomials. In addition, for the classical
random matrices (Gaussian, Wishart, and multivariate beta matrices), we analyze
the limiting behavior of the EEC approximation as the matrix size goes to
infinity under the so-called edge-asymptotic normalization. It is shown that
the limit of the EEC formula approximates well the Tracy-Widom distributions in
the upper tail area, as does the EEC formula when the matrix size is finite.Comment: 30 pages, 3 figures, 3 table
Skewness and kurtosis as locally best invariant tests of normality
Consider testing normality against a one-parameter family of univariate
distributions containing the normal distribution as the boundary, e.g., the
family of -distributions or an infinitely divisible family with finite
variance. We prove that under mild regularity conditions, the sample skewness
is the locally best invariant (LBI) test of normality against a wide class of
asymmetric families and the kurtosis is the LBI test against symmetric
families. We also discuss non-regular cases such as testing normality against
the stable family and some related results in the multivariate cases
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